Question

Daca log in baza 40 din 100=a, calculati login baza 20 din 50.

Answer 1

$a = \log_{40}100= \log_{40}(2\cdot 50) = \log_{40}2 + \log_{40}50 = \\ \\ = \log_{40}2 + \dfrac{1}{\log_{50}40} = \log_{40}2 +\dfrac{1}{\log_{50}2\cdot 20} = \\ \\ = \log_{40}2 +\dfrac{1}{\log_{50}2+\log_{50}20} = \log_{40}2 +\dfrac{1}{\log_{50}2+\dfrac{1}{\log_{20}50}} \\ \\ \Rightarrow \log_{50}2+\dfrac{1}{\log_{20}50} = \dfrac{1}{a- \log_{40}2} \Rightarrow \dfrac{1}{\log_{20}50} = \dfrac{1}{a- \log_{40}2} - \log_{50}2 \Rightarrow \\ \\ \Rightarrow \log_{20}50 = \dfrac{1}{\dfrac{1}{a- \log_{40}2} - \log_{50}2}$